Tropical Geometry and Five Dimensional Higgs Branches at Infinite Coupling

TL;DR

This paper addresses the problem of characterizing the Higgs branch ${\cal H}_\infty$ of a three-parameter family of 5d ${\cal N}=1$ SQCD theories at infinite coupling, revealing a rich structure with up to three components. It develops a novel conjecture that reads the quiver data directly from maximally subdivided brane webs, employing tropical geometry via stable intersections and incorporating 7-brane contributions to compute edges between quiver nodes. The main results provide a complete classification of ${\cal H}_\infty$ across four regional regimes in $(N_c,N_f,k)$, describing components as spaces of dressed monopole operators and detailing their global symmetry, dimensions, and pairwise/triple intersections. The findings demonstrate that the Higgs branch at infinite coupling is a union of hyperKähler cones, each described by a 3d Coulomb branch, and establish a broadly applicable framework for analyzing such moduli spaces in higher-dimensional gauge theories with brane realizations.

Abstract

Superconformal five dimensional theories have a rich structure of phases and brane webs play a crucial role in studying their properties. This paper is devoted to the study of a three parameter family of SQCD theories, given by the number of colors $N_c$ for an $SU(N_c)$ gauge theory, number of fundamental flavors $N_f$, and the Chern Simons level $k$. The study of their infinite coupling Higgs branch is a long standing problem and reveals a rich pattern of moduli spaces, depending on the 3 values in a critical way. For a generic choice of the parameters we find a surprising number of 3 different components, with intersections that are closures of height 2 nilpotent orbits of the flavor symmetry. This is in contrast to previous studies where except for one case ($N_c=2, N_f=2$), the parameters were restricted to the cases of Higgs branches that have only one component. The new feature is achieved thanks to a concept in tropical geometry which is called stable intersection and allows for a computation of the Higgs branch to almost all the cases which were previously unknown for this three parameter family apart form certain small number of exceptional theories with low rank gauge group. A crucial feature in the construction of the Higgs branch is the notion of dressed monopole operators.

Figures (55)

• Figure 1: Toric diagram corresponding to the 5d SQCD theory with SU(2) gauge group and $N_f=2$.
• Figure 2: Five brane webs (dual to the toric diagram in figure \ref{['fig:E3toric']}) corresponding to the 5d SQCD theory with SU(2) gauge group and $N_f=2$ before and after taking the gauge coupling $g$ to infinity and all the masses $m_i$ as well as the VEV $a$ of the adjoint scalar field to zero. The horizontal lines represent D5-branes, the vertical lines represent NS5-branes and the diagonal lines represent $(1,-1)$ five branes. Each circle represents a seven brane of the same type as the five brane that ends on it.
• Figure 3: Two distinct phases of brane webs, representing two different components of the Higgs branch of $SU(2)$ with $N_f=2$ at infinite coupling. (a) Corresponds to a component of the Higgs branch with quaternionic dimension $d_{\mathbb{H}}=3-1=2$. (b) Corresponds to a component of the Higgs branch with quaternionic dimension $d_{\mathbb{H}}=2-1=1$. Compare with Figure 30 of Franco:2005zu.
• Figure 4: Stable intersection between two tropical curves. Left: the original tropical curves. Center: a small deformation that moves the curves away from each other. Right: dual toric diagram. The stable intersection is the sum over the areas of polygons in the toric diagram that have edges of both colours. In this case is $1+1=2$.
• Figure 5: Toric diagram corresponding to the 5d $\mathcal{N}=1$ SQCD theory with SU(2) gauge group and $N_f=3$.

Theorems & Definitions (1)

• Conjecture 1